Top subcategories

... mathematics. A first-order theory consists of a set of axioms (usually finite or recursively
enumerable) and the statements deducible from them.
Peano arithmetic is a first-order theory commonly formalized independently in first-order
logic. It constitutes a fundamental formalism for arithmetic, and ...

... 1. Continue defining and exploring first-order theory of simple arithmetic, iQ.
i Q is a first-order finite axiomatization of a “number-like” domain. Even though i Q
is extremely weak as you see from Problem Set 3 from Boolos & Jeffrey, we can,
nevertheless, show in constructive type theory, either ...

... • Used to join atomic symbols to form complex structure
• Valid connectives are as follows
i)Not or negation  Denoted by ̚  if P is true then ̚ P is false
ii)Conjunction Denoted by AND/˄ P˄Q will be true if both of
them is true
iii)Disjunction Denoted by OR/˅ P˅Q will be true if only one of
th ...

... So we get: PA, except that axioms assert only the
existence of finite sets definable with
formulas
(formulas with no string-quantifiers and with bounded
number-quantifiers.)
Such formulas correspond to a (weak) complexity class:
constant-depth Boolean circuits of polynomial-size
(aka AC0). Denote th ...

... • Atomic sentences make claims that have
truth value. In other words, they are TRUE
or FALSE.
• An atomic sentence consists of a predicate
followed by a list of names, the number of
which correspond to the predicate’s arity.
• Names refer to objects. Predicates refer to
properties or relations of ob ...

... Gödel then continues to construct a lot of relations, which he proves are recursive. These are for
example P(x,y): “x is the Gödel number of a proof of the wf with Gödel number y”
He then construct the wf U that together with a relation W says something like “This wf can’t be
proven” when it’s anal ...

... In any logical language expressive enough
to describe the properties of the natural
numbers, there are true statements that are
undecidable -- their truth cannot be
established by any algorithm.
...

Principia Mathematica

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.